**Additive Chernoff**

Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.

\begin{gather*}
\operatorname{Pr}\left(\frac{1}{n} \sum X_i \geq p+\varepsilon\right) \leq\left(\left(\frac{p}{p+\varepsilon}\right)^{p+\varepsilon}\left(\frac{1-p}{1-p-\varepsilon}\right)^{1-p-\varepsilon}\right)^n=e^{-D(p+\varepsilon \mathbin\| p) n }
\\
\operatorname{Pr} \left(\frac{1}{n} \sum X_i \leq p-\varepsilon\right) \leq\left(\left(\frac{p}{p-\varepsilon}\right)^{p-\varepsilon}\left(\frac{1-p}{1-p+\varepsilon}\right)^{1-p+\varepsilon}\right)^n=e^{-D(p-\varepsilon \mathbin\| p) n}
\end{gather*}

where
$$
D(x \mathbin\| y)=x \ln \frac{x}{y}+(1-x) \ln \left(\frac{1-x}{1-y}\right).
$$

 1. For sums of i.i.d random variables, is there any bound tighter than the additive Chernoff?
 
  [1]: https://cstheory.stackexchange.com/questions/53412/additive-chernoff-bound